Phy-104 Electricity and Magnetism

1. Mother and daughter are both enjoying the effects of electrically charging their bodies. Each individual hair on their heads becomes charged and exerts a repulsive force on the other hairs, resulting in the �stand-up�� hairdos that you see here. (Courtesy of Resonance Research Corporation)Mother and daughter are both enjoying the effects of electrically charging their bodies. Each individual hair on their heads becomes charged and exerts a repulsive force on the other hairs, resulting in the �stand-up�� hairdos that you see here. (Courtesy of Resonance Research Corporation)

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6. Figure 23.1 (a) A negatively charged rubber rod suspended by a thread is attracted to a positively charged glass rod. (b) A negatively charged rubber rod is repelled by another negatively charged rubber rod.Figure 23.1 (a) A negatively charged rubber rod suspended by a thread is attracted to a positively charged glass rod. (b) A negatively charged rubber rod is repelled by another negatively charged rubber rod.

7. Figure 23.2 When a glass rod is rubbed with silk, electrons are transferred from the glass to the silk. Because of conservation of charge, each electron adds negative charge to the silk, and an equal positive charge is left behind on the rod. Also, because the charges are transferred in discrete bundles, the charges on the two objects are +/-e, or +/-2e, or +/-3e, and so on.Figure 23.2 When a glass rod is rubbed with silk, electrons are transferred from the glass to the silk. Because of conservation of charge, each electron adds negative charge to the silk, and an equal positive charge is left behind on the rod. Also, because the charges are transferred in discrete bundles, the charges on the two objects are +/-e, or +/-2e, or +/-3e, and so on.

8. Figure 23.4 Charging a metallic object by induction (that is, the two objects never touch each other). (a) A neutral metallic sphere, with equal numbers of positive and negative charges. (b) The electrons on the neutral sphere are redistributed when a charged rubber rod is placed near the sphere. (c) When the sphere is grounded, some of its electrons leave through the ground wire. (d) When the ground connection is removed, the sphere has excess positive charge that is nonuniformly distributed. (e) When the rod is removed, the remaining electrons redistribute uniformly and there is a net uniform distribution of positive charge on the sphere.Figure 23.4 Charging a metallic object by induction (that is, the two objects never touch each other). (a) A neutral metallic sphere, with equal numbers of positive and negative charges. (b) The electrons on the neutral sphere are redistributed when a charged rubber rod is placed near the sphere. (c) When the sphere is grounded, some of its electrons leave through the ground wire. (d) When the ground connection is removed, the sphere has excess positive charge that is nonuniformly distributed. (e) When the rod is removed, the remaining electrons redistribute uniformly and there is a net uniform distribution of positive charge on the sphere.

9. Figure 23.5 (a) The charged object on the left induces a charge distribution on the surface of an insulator due to realignment of charges in the molecules. Figure 23.5 (a) The charged object on the left induces a charge distribution on the surface of an insulator due to realignment of charges in the molecules.

10. Figure 23.5 (b) A charged comb attracts bits of paper because charges in molecules in the paper are realigned. Figure 23.5 (b) A charged comb attracts bits of paper because charges in molecules in the paper are realigned.

11. Figure 23.6 Coulomb�s torsion balance, used to establish the inverse-square law for the electric force between two charges.Figure 23.6 Coulomb�s torsion balance, used to establish the inverse-square law for the electric force between two charges.

13. Figure 23.7 Two point charges separated by a distance r exert a force on each other that is given by Coulomb�s law. The force F21 exerted by q2 on q1 is equal in magnitude and opposite in direction to the force F12 exerted by q1 on q2. (a) When the charges are of the same sign, the force is repulsive. (b) When the charges are of opposite signs, the force is attractive.Figure 23.7 Two point charges separated by a distance r exert a force on each other that is given by Coulomb�s law. The force F21 exerted by q2 on q1 is equal in magnitude and opposite in direction to the force F12 exerted by q1 on q2. (a) When the charges are of the same sign, the force is repulsive. (b) When the charges are of opposite signs, the force is attractive.

14. Figure 23.7 Two point charges separated by a distance r exert a force on each other that is given by Coulomb�s law. The force F21 exerted by q2 on q1 is equal in magnitude and opposite in direction to the force F12 exerted by q1 on q2. (a) When the charges are of the same sign, the force is repulsive. Figure 23.7 Two point charges separated by a distance r exert a force on each other that is given by Coulomb�s law. The force F21 exerted by q2 on q1 is equal in magnitude and opposite in direction to the force F12 exerted by q1 on q2. (a) When the charges are of the same sign, the force is repulsive.

15. Figure 23.8 (Example 23.2) The force exerted by q1 on q3 is F13. The force exerted by q2 on q3 is F23. The resultant force F3 exerted on q3 is the vector sum F13 F23.Figure 23.8 (Example 23.2) The force exerted by q1 on q3 is F13. The force exerted by q2 on q3 is F23. The resultant force F3 exerted on q3 is the vector sum F13 F23.

16. Figure 23.9 (Example 23.3) Three point charges are placed along the x axis. If the resultant force acting on q3 is zero, then the force F13 exerted by q1 on q3 must be equal in magnitude and opposite in direction to the force F23 exerted by q2 on q3.Figure 23.9 (Example 23.3) Three point charges are placed along the x axis. If the resultant force acting on q3 is zero, then the force F13 exerted by q1 on q3 must be equal in magnitude and opposite in direction to the force F23 exerted by q2 on q3.

17. Figure 23.11 A small positive test charge q0 placed near an object carrying a much larger positive charge Q experiences an electric field E directed as shown.Figure 23.11 A small positive test charge q0 placed near an object carrying a much larger positive charge Q experiences an electric field E directed as shown.

18. Figure 23.12 (a) For a small enough test charge q0, the charge distribution on the sphere is undisturbed. (b) When the test charge q0is greater, the charge distribution on the sphere is disturbed as the result of the proximity of q0.Figure 23.12 (a) For a small enough test charge q0, the charge distribution on the sphere is undisturbed. (b) When the test charge q0is greater, the charge distribution on the sphere is disturbed as the result of the proximity of q0.

19. where r � is a unit vector directed from q toward q0. This force in Figure 23.13a is Active Figure 23.13 A test charge q0 at point P is a distance r from a point charge q. (a) If q is positive, then the force on the test charge is directed away from q. (b) For the positive source charge, the electric field at P points radially outward from q. (c) If q is negative, then the force on the test charge is directed toward q. (d) For the negative source charge, the electric field at P points radially inward toward q.where r � is a unit vector directed from q toward q0. This force in Figure 23.13a is Active Figure 23.13 A test charge q0 at point P is a distance r from a point charge q. (a) If q is positive, then the force on the test charge is directed away from q. (b) For the positive source charge, the electric field at P points radially outward from q. (c) If q is negative, then the force on the test charge is directed toward q. (d) For the negative source charge, the electric field at P points radially inward toward q.

20. Figure 23.15 (Example 23.6) The total electric field E at P due to two charges of equal magnitude and opposite sign (an electric dipole) equals the vector sum E1 + E2. The field E1 is due to the positive charge q, and E2 is the field due to the negative charge q.Figure 23.15 (Example 23.6) The total electric field E at P due to two charges of equal magnitude and opposite sign (an electric dipole) equals the vector sum E1 + E2. The field E1 is due to the positive charge q, and E2 is the field due to the negative charge q.

21. Figure 23.20 Electric field lines penetrating two surfaces. The magnitude of the field is greater on surface A than on surface B.Figure 23.20 Electric field lines penetrating two surfaces. The magnitude of the field is greater on surface A than on surface B.

22. Figure 23.21 The electric field lines for a point charge. (a) For a positive point charge, the lines are directed radially outward. (b) For a negative point charge, the lines are directed radially inward. Note that the figures show only those field lines that lie in the plane of the page. (c) The dark areas are small pieces of thread suspended in oil, which align with the electric field produced by a small charged conductor at the center.Figure 23.21 The electric field lines for a point charge. (a) For a positive point charge, the lines are directed radially outward. (b) For a negative point charge, the lines are directed radially inward. Note that the figures show only those field lines that lie in the plane of the page. (c) The dark areas are small pieces of thread suspended in oil, which align with the electric field produced by a small charged conductor at the center.

23. Figure 23.21 The electric field lines for a point charge. (a) For a positive point charge, the lines are directed radially outward. Figure 23.21 The electric field lines for a point charge. (a) For a positive point charge, the lines are directed radially outward.

24. Figure 23.21 The electric field lines for a point charge. (b) For a negative point charge, the lines are directed radially inward. Note that the figures show only those field lines that lie in the plane of the page. Figure 23.21 The electric field lines for a point charge. (b) For a negative point charge, the lines are directed radially inward. Note that the figures show only those field lines that lie in the plane of the page.

25. Figure 23.21 The electric field lines for a point charge. (c) The dark areas are small pieces of thread suspended in oil, which align with the electric field produced by a small charged conductor at the center. Figure 23.21 The electric field lines for a point charge. (c) The dark areas are small pieces of thread suspended in oil, which align with the electric field produced by a small charged conductor at the center.

26. Figure 23.22 (a) The electric field lines for two point charges of equal magnitude and opposite sign (an electric dipole). The number of lines leaving the positive charge equals the number terminating at the negative charge. (b) The dark lines are small pieces of thread suspended in oil, which align with the electric field of a dipole.Figure 23.22 (a) The electric field lines for two point charges of equal magnitude and opposite sign (an electric dipole). The number of lines leaving the positive charge equals the number terminating at the negative charge. (b) The dark lines are small pieces of thread suspended in oil, which align with the electric field of a dipole.

27. Figure 23.22 (b) The dark lines are small pieces of thread suspended in oil, which align with the electric field of a dipole.Figure 23.22 (b) The dark lines are small pieces of thread suspended in oil, which align with the electric field of a dipole.

28. Figure 23.23 (a) The electric field lines for two positive point charges. (The locations A, B, and C are discussed in Quick Quiz 23.7.)Figure 23.23 (a) The electric field lines for two positive point charges. (The locations A, B, and C are discussed in Quick Quiz 23.7.)

29. Figure 23.23 (b) Pieces of thread suspended in oil, which align with the electric field created by two equal-magnitude positive charges. Courtesy of Harold M. Waage, Princeton University Figure 23.23 (b) Pieces of thread suspended in oil, which align with the electric field created by two equal-magnitude positive charges. Courtesy of Harold M. Waage, Princeton University

30. Active Figure 23.24 The electric field lines for a point charge +2q and a second point charge -q. Note that two lines leave +2q for every one that terminates on -q.Active Figure 23.24 The electric field lines for a point charge +2q and a second point charge -q. Note that two lines leave +2q for every one that terminates on -q.

31. Figure 23.25 (Example 23.10) A positive point charge q in a uniform electric field E undergoes constant acceleration in the direction of the field.Figure 23.25 (Example 23.10) A positive point charge q in a uniform electric field E undergoes constant acceleration in the direction of the field.

32. E (x, y) Figure 23.26 An electron is projected horizontally into a uniform electric field produced by two charged plates. The electron undergoes a downward acceleration (opposite E), and its motion is parabolic while it is between the plates.E (x, y) Figure 23.26 An electron is projected horizontally into a uniform electric field produced by two charged plates. The electron undergoes a downward acceleration (opposite E), and its motion is parabolic while it is between the plates.

33. Figure 23.27 Schematic diagram of a cathode ray tube. Electrons leaving the cathode C are accelerated to the anode A. In addition to accelerating electrons, the electron gun is also used to focus the beam of electrons, and the plates deflect the beam.Figure 23.27 Schematic diagram of a cathode ray tube. Electrons leaving the cathode C are accelerated to the anode A. In addition to accelerating electrons, the electron gun is also used to focus the beam of electrons, and the plates deflect the beam.

39. Fig 24-CO, p.737 In a table-top plasma ball, the colorful lines emanating from the sphere give evidence of strong electric fields. Using Gauss�s law, we show in this chapter that the electric field surrounding a charged sphere is identical to that of a point charge. (Getty Images)Fig 24-CO, p.737 In a table-top plasma ball, the colorful lines emanating from the sphere give evidence of strong electric fields. Using Gauss�s law, we show in this chapter that the electric field surrounding a charged sphere is identical to that of a point charge. (Getty Images)

40. 24-1 ????? (??????) ??????? : - ???? ????? ??????? (F) ???? ???? ?????? ??????? ???? ???? ?? ??? ?? ?????? A. - F ???? ????? ???? ??? ????? ????? (A) ???? ?????? ??????? (E) ??????? ??? ?????. + +

41. ??? ??? ?????? ??????? ???? ????? ( ???? ????? ????????) ???: Figure 24.1 Field lines representing a uniform electric field penetrating a plane of area A perpendicular to the field. The electric flux ?E hrough this area is equal to EA. Figure 24.1 Field lines representing a uniform electric field penetrating a plane of area A perpendicular to the field. The electric flux ?E hrough this area is equal to EA.

42. ????? ???? ?????? ??? ?????? ??? ????? ????? ??????? ??? ???? ?????? ???????? ??? ????? ????? Figure 24.2 Field lines representing a uniform electric field penetrating an area A� that is at an angle ? to the field. Because the number of lines that go through the area A is the same as the number that go through A, the flux through A is equal to the flux through A and is given by ?E= EAcos ?.Figure 24.2 Field lines representing a uniform electric field penetrating an area A� that is at an angle ? to the field. Because the number of lines that go through the area A is the same as the number that go through A, the flux through A is equal to the flux through A and is given by ?E= EAcos ?.

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44. Figure 24.3 A small element of surface area Ai. The electric field makes an angle ?i with he vector ?Ai, defined as being normal to the surface element, and the flux through the element is equal to Ei ?Ai cos ?i . Figure 24.3 A small element of surface area Ai. The electric field makes an angle ?i with he vector ?Ai, defined as being normal to the surface element, and the flux through the element is equal to Ei ?Ai cos ?i .

45. Karl Friedrich Gauss German mathematician and astronomer (1777�1855) Gauss received a doctoral degree in mathematics from the University of Helmstedt in 1799. In addition to his work in electromagnetism, he made contributions to mathematics and science in number theory, statistics, non-Euclidean geometry, and cometary orbital mechanics. He was a founder of the German Magnetic Union, which studies the Earth�s magnetic field on a continual basis.Karl Friedrich Gauss German mathematician and astronomer (1777�1855) Gauss received a doctoral degree in mathematics from the University of Helmstedt in 1799. In addition to his work in electromagnetism, he made contributions to mathematics and science in number theory, statistics, non-Euclidean geometry, and cometary orbital mechanics. He was a founder of the German Magnetic Union, which studies the Earth�s magnetic field on a continual basis.

46. ???? ????? ?????? ??? ??? ????? ???? ???? ?????? ??? ?????? ??????? ??? ??? ?????? ????? ???? ?????? ???? ???? ?????? ??????: Active Figure 24.4 A closed surface in an electric field. The area vectors ?Ai are, by convention,normal to the surface and point outward. The flux through an area element can be positive (element 1), zero (element 2), or negative (element 3). Active Figure 24.4 A closed surface in an electric field. The area vectors ?Ai are, by convention,normal to the surface and point outward. The flux through an area element can be positive (element 1), zero (element 2), or negative (element 3).

47. ???? 24-1 ? 686 ???? ???? ????? ??????? ????? ???? ??? ???? ??? ???? ??? ?? ???? ????? ?????? ?????? Figure 24.5 (Example 24.2) A closed surface in the shape of a cube in a uniform electric field oriented parallel to the x axis. Side 4 is the bottom of the cube, and side 1 is opposite side 2.Figure 24.5 (Example 24.2) A closed surface in the shape of a cube in a uniform electric field oriented parallel to the x axis. Side 4 is the bottom of the cube, and side 1 is opposite side 2.

48. 24-2: ????? ????? ??? ?????? ??? ???? ???? ????? ???????? ????? ????? ??????? ??? ?????..... and from Equation 24.4 we find that the net flux through the gaussian surface is E EdA E dA E dA Figure 24.6 A spherical gaussian surface of radius r surrounding a point charge q. When the charge is at the center of the sphere, the electric field is everywhere normal to the surface and constant in magnitude.and from Equation 24.4 we find that the net flux through the gaussian surface is E EdA E dA E dA Figure 24.6 A spherical gaussian surface of radius r surrounding a point charge q. When the charge is at the center of the sphere, the electric field is everywhere normal to the surface and constant in magnitude.

49. Figure 24.7 Closed surfaces of various shapes surrounding a charge q. The net electric flux is the same through all surfaces.Figure 24.7 Closed surfaces of various shapes surrounding a charge q. The net electric flux is the same through all surfaces.

50. Figure 24.8 A point charge located outside a closed surface. The number of lines entering the surface equals the number leaving the surface.Figure 24.8 A point charge located outside a closed surface. The number of lines entering the surface equals the number leaving the surface.

51. ???? ????? ?????: ??? ?? ?? ??? ?????? ??? ??? ????? (????? ??????) ??????? ??? ?? ???? ??? ?????? ??????? � ??????? ????????_ ??? ??????? ?????.

52. Active Figure 24.9 The net electric flux through any closed surface depends only on the charge inside that surface. The net flux through surface S is q1 /0 , the net flux through surface Sis (q 2 + q3)/e 0 , and the net flux through surface S is zero. Charge q4 does not contribute to the flux through any surface because it is outside all surfaces.Active Figure 24.9 The net electric flux through any closed surface depends only on the charge inside that surface. The net flux through surface S is q1 /0 , the net flux through surface Sis (q 2 + q3)/e 0 , and the net flux through surface S is zero. Charge q4 does not contribute to the flux through any surface because it is outside all surfaces.

53. 24-3 ??????? ??? ????? ????? 1- ???? ??? ?????? ??????? ??? ???? ?????:

54. Figure 24.10 (Example 24.4) The point charge q is at the center of the spherical gaussian surface, and E is parallel to d A at every point on the surface.Figure 24.10 (Example 24.4) The point charge q is at the center of the spherical gaussian surface, and E is parallel to d A at every point on the surface.

55. ?????? ?????? q ? ??????? ??????? ?????? Figure 24.11 (Example 24.5) A uniformly charged insulating sphere of radius a and total charge Q. (a) For points outside the sphere, a large spherical gaussian surface is drawn concentric with the sphere. In diagrams such as this, the dotted line represents the intersection of the gaussian surface with the plane of the page. (b) For points inside the sphere, a spherical gaussian surface smaller than the sphere is drawn.Figure 24.11 (Example 24.5) A uniformly charged insulating sphere of radius a and total charge Q. (a) For points outside the sphere, a large spherical gaussian surface is drawn concentric with the sphere. In diagrams such as this, the dotted line represents the intersection of the gaussian surface with the plane of the page. (b) For points inside the sphere, a spherical gaussian surface smaller than the sphere is drawn.

56. Figure 24.12 (Example 24.5) A plot of E versus r for a uniformly charged insulating phere. The electric field inside the sphere (r? a) varies linearly with r. The field outside the sphere (r? a) is the same as that of a point charge Q located at r 0. Figure 24.12 (Example 24.5) A plot of E versus r for a uniformly charged insulating phere. The electric field inside the sphere (r? a) varies linearly with r. The field outside the sphere (r? a) is the same as that of a point charge Q located at r 0.

57. Figure 24.13 (Example 24.6) (a) The electric field inside a uniformly charged spherical shell is zero. The field outside is the same as that due to a point charge Q located at the center of the shell. (b) Gaussian surface for r? a. (c) Gaussian surface for r? a. Figure 24.13 (Example 24.6) (a) The electric field inside a uniformly charged spherical shell is zero. The field outside is the same as that due to a point charge Q located at the center of the shell. (b) Gaussian surface for r? a. (c) Gaussian surface for r? a.

58. Figure 24.13 (Example 24.6) (a) The electric field inside a uniformly charged spherical shell is zero. The field outside is the same as that due to a point charge Q located at the center of the shell. Figure 24.13 (Example 24.6) (a) The electric field inside a uniformly charged spherical shell is zero. The field outside is the same as that due to a point charge Q located at the center of the shell.

59. ?????? ??????? ?????? ????? ?????? ???? ????? ???????? ??? ????? ?????? Figure 24.13 (Example 24.6) (b) Gaussian surface for r? a. Figure 24.13 (Example 24.6) (b) Gaussian surface for r? a.

60. Figure 24.13 (Example 24.6) (c) Gaussian surface for r? a. Figure 24.13 (Example 24.6) (c) Gaussian surface for r? a.

61. 4- ???????? ??????? ?????? ?????? ????? ??????? Figure 24.14 (Example 24.7) (a) An infinite line of charge surrounded by a cylindrical gaussian surface concentric with the line. (b) An end view shows that the electric field at the cylindrical surface is constant in magnitude and perpendicular to the surface.Figure 24.14 (Example 24.7) (a) An infinite line of charge surrounded by a cylindrical gaussian surface concentric with the line. (b) An end view shows that the electric field at the cylindrical surface is constant in magnitude and perpendicular to the surface.

62. Figure 24.14 (Example 24.7) (a) An infinite line of charge surrounded by a cylindrical gaussian surface concentric with the line. Figure 24.14 (Example 24.7) (a) An infinite line of charge surrounded by a cylindrical gaussian surface concentric with the line.

63. Figure 24.14 (Example 24.7) (b) An end view shows that the electric field at the cylindrical surface is constant in magnitude and perpendicular to the surface. Figure 24.14 (Example 24.7) (b) An end view shows that the electric field at the cylindrical surface is constant in magnitude and perpendicular to the surface.

64. Figure 24.15 (Example 24.8) A cylindrical gaussian surface penetrating an infinite plane of charge. The flux is Ea through each end of the gaussian surface and zero through its curved surface.Figure 24.15 (Example 24.8) A cylindrical gaussian surface penetrating an infinite plane of charge. The flux is Ea through each end of the gaussian surface and zero through its curved surface.

66. ?????? ??? ??????? ????????? ????????? ???????? ??????????: ???? ??? ????? ?????? ????? ??????? ??????? ??? ???????? ????? ??????? ???????

67. 24.4: ???????? ?? ???? ?????? ????????? * ?????? ???? ??????? ??????? ????? ??? ????? ???? ????? ???? ????? ?????? ???? ?????? ?????? ????? ??? ????? ?????? ??? ????? ???? ????? ????? ??? ?????? ???? ?? ???? ????? ??????????? ?????? ???????? ??????: ??? ?????? ???? ?????? ????? ???? ????? ??????? ??? ????? ??????? ?????? ???????? ????? ????? ??????? ??????? � ????? ????? ?????- ??? ??????? ??????? ??? ???? ?? ??? ?????? ?????? ????? ??? ?????? ???? ?????? ???? ?????? ????? ????? ??? ????? ??? ???? ?????? ????? ??? � 0E = s/e � ??? ????? ?????? ??????? ???? ????? ?? ???? ??? ????

68. ??? ??? ???? ???? ???? ????? ?????? ????? ??????? ??? ???? ?????? ( ??? ?? ????) ?? ???? ??????? ??????? ?????? ???? ???? ????? ???? ????? ????? ?????? ???????? ???? ???? ????? ?????? ???? ?????? ????? ???. Figure 24.16 A conducting slab in an external electric field E. The charges induced on the two surfaces of the slab produce an electric field that opposes the external field, giving a resultant field of zero inside the slab.Figure 24.16 A conducting slab in an external electric field E. The charges induced on the two surfaces of the slab produce an electric field that opposes the external field, giving a resultant field of zero inside the slab.

69. Figure 24.17 A conductor of arbitrary shape. The broken line represents a gaussian surface just inside the surface of the conductor.Figure 24.17 A conductor of arbitrary shape. The broken line represents a gaussian surface just inside the surface of the conductor.

70. Figure 24-18 A gaussian surface in the shape of a small cylinder is used to calculate the electric field just outside a charged conductor. The flux through the gaussian surface is EA. Remember that E is zero inside the conductor.Figure 24-18 A gaussian surface in the shape of a small cylinder is used to calculate the electric field just outside a charged conductor. The flux through the gaussian surface is EA. Remember that E is zero inside the conductor.

71. Figure 24.19 Electric field pattern surrounding a charged conducting plate placed near an oppositely charged conducting cylinder. Small pieces of thread suspended in oil align with the electric field lines. Note that (1) the field lines are perpendicular to both conductors and (2) there are no lines inside the cylinder (E= 0).Figure 24.19 Electric field pattern surrounding a charged conducting plate placed near an oppositely charged conducting cylinder. Small pieces of thread suspended in oil align with the electric field lines. Note that (1) the field lines are perpendicular to both conductors and (2) there are no lines inside the cylinder (E= 0).

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73. Figure 24.21 (Example 24.10) A plot of E versus r for the two-conductor system shown in Figure 24.20.Figure 24.21 (Example 24.10) A plot of E versus r for the two-conductor system shown in Figure 24.20.

75. Figure 24.23 The area element A subtends a solid angle ??= (?A cos ?)/r 2 at the charge q. Figure 24.23 The area element A subtends a solid angle ??= (?A cos ?)/r 2 at the charge q.